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MP Keith Vaz, who raised questions over Baroness Butler-Sloss' appointment to the abuse inquiry last week, has said the process has become "shambolic".
The Chair of the Home Affairs Select Committee said he was "not surprised" by Lady Butler-Sloss' decision to stand down, adding that "it is the right one".
"As I pointed out to [Home Office permanent secretary Mark] Sedwill, the public would be concerned that a member of Parliament, no matter how distinguished, had been appointed to head this important panel.
"The whole inquiry process is becoming shambolic: missing files, ministers refusing to read reports and now the chair resigning before the inquiry has even commenced."
Downing Street says Baroness Butler-Sloss' stepping down from the abuse inquiry was "entirely her decision".
Home Secretary Theresa May has said she is "deeply saddened" but understands and respects Baroness Butler-Sloss's decision to stand down as the chair of an inquiry into allegations of historic child sex abuse within the establishment.
Former judge Baroness Butler-Sloss is to step down from her role leading an inquiry into alleged child abuse, Downing Street has announced.
Shadow Home Secretary Yvette Cooper has repeatedly dodged questions over whether Labour backed Lady Butler-Sloss as the right person to chair an inquiry into allegations of a child sex abuse cover-up.
Ms Cooper told Murnaghan on Sky News: "I think she is an extremely experienced person who will be very good to do this job but she also needs the right people around her, she needs the Home Office to take action to make sure they address all of these concerns.
"If they can't they will need to make changes and rethink the whole thing but I think the ball should be in the Home Office's court now to set this up in the right way and to make sure they can do that because I do think she has immense expertise that should be drawn upon."
The Archbishop of Canterbury says he expects more allegations of child abuse within the Church of England to come to the fore as a wide-ranging inquest gets underway.
Asked on BBC's The Andrew Marr Show if he expected more such stories in the wake of Baroness Butler-Sloss' probe, Justin Welby said: "It is becoming clearer and clearer that for many, many years, things were not dealt with as they should have been dealt with.
"We must show justice to survivors of abuse - that is the first and absolute principal," he said.
The Home Office again has insisted it stands by the appointment of Baroness Butler-Sloss "unreservedly", amid claims made by The Times (£) that she refused to go public about a bishop implicated in a scandal. A spokesman said:
The Home Office has again been forced to defend the appointment of Baroness Butler-Sloss to run an inquiry into allegations of an establishment cover-up of child abuse amid claims she refused to go public about a bishop implicated in a scandal.
According to The Times (£), the retired judge told a victim of alleged abuse that she did not want the claims to be in the public domain because "the press would love a bishop".
Her comments were reportedly made in 2011 during a meeting at the House of Lords with Phil Johnson, who was abused by priests when he was a choirboy, the newspaper said.
In a statement, Lady Butler-Sloss said she had always "striven to be fair and compassionate", towards victims of crime. She added: "I have never put the reputation of any institution, including the Church of England above the pursuit of justice for victims".
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The Baroness says she did not consider the "difficulties" caused by her background and family connections to the child sex abuse inquiry.
Questions have been raised of former High Court judge Baroness Butler-Sloss since her appointment last week.